Time-dependent P\'olya urn

We consider a time-dependent version of a P\'olya urn containing black and white balls. At each time $n$ a ball is drawn from the urn at random and replaced in the urn along with $\sigma_n$ additional balls of the same colour. The proportion of white balls converges almost surely to a random limit $\Theta$, and $\mathcal{D}=\{\Theta\in\{0,1\}\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $(\sigma_n)$, between the regimes ${\mathbb P}(\mathcal{D})=1$ and ${\mathbb P}(\mathcal{D})<1$ was obtained by R. Pemantle in 1990. We describe the phase transition between the cases ${\mathbb P}(\mathcal{D})=0$ and ${\mathbb P}(\mathcal{D})>0$. Further, we study the stronger monopoly event $\mathcal{M}$ when one of the colours eventually stops reappearing, and analyse the phase transition between the regimes ${\mathbb P}(\mathcal{M})=0$, ${\mathbb P}(\mathcal{M})\in (0,1)$, and ${\mathbb P}(\mathcal{M})=1$.